GTP Workshop on Modeling MHD Turbulence; Applications to Planetary and Stellar dynamos

Navier-Stokes as well as magnetohydrodynamic (MHD) turbulence has long been described in terms of energy spectra and cascades of ideal quadratic invariants such as energy or enstrophy. The concept of energy transfer between eddies is closely related to the energy cascade picture and a number of numerical studies have been devoted to the characterization of nonlinear energy exchanges between different scales in fully developed turbulence. To the best of our knowledge, former studies have focused on isotropic turbulence [1, 2, 3, 4, 5]. In that case, the Fourier representation of the velocity and magnetic field variables (ui(~k) and bi(~k)) is commonly adopted and the Fourier space is often split into shells of wave vectors sn that contain all the wave vectors satisfying kn ≤ ||~k| | < kn+1. The study of energy exchanges between the modes that belong to these shells allows to determine the direction of cascade as well as the intensity of energy fluxes [6, 7, 8, 9, 10]. In MHD, especially in applications to planetary and stellar dynamos, turbulence can however rarely be considered as isotropic. Indeed, external magnetic fields and rotation definitively impose an anisotropy direction and have a strong influence on the energy cascade. For instance, it is well known that rotation tends to inhibit the energy cascade towards the small scales. Similarly, a strong magnetic field tends to produce quasi two-dimensional turbulence in which the cascade mechanisms are significantly different from thos